Open Access
VOL. 57 | 2010 A survey of Ricci curvature for metric spaces and Markov chains
Yann Ollivier

Editor(s) Motoko Kotani, Masanori Hino, Takashi Kumagai

Adv. Stud. Pure Math., 2010: 343-381 (2010) DOI: 10.2969/aspm/05710343


This text is a presentation of the general context and results of [Oll07] and [Oll09], with comments on related work. The goal is to present a notion of Ricci curvature valid on arbitrary metric spaces, such as graphs, and to generalize a series of classical theorems in positive Ricci curvature, such as spectral gap estimates, concentration of measure or log-Sobolev inequalities.

The necessary background (concentration of measure, curvature in Riemannian geometry, convergence of Markov chains) is covered in the first section. Special emphasis is put on open questions of varying difficulty.


Published: 1 January 2010
First available in Project Euclid: 24 November 2018

zbMATH: 1204.53035
MathSciNet: MR2648269

Digital Object Identifier: 10.2969/aspm/05710343

Primary: 51F99 , 53B21 , 60B99

Keywords: concentration of measure , Markov chains , metric measure spaces , Ricci curvature , Wasserstein distances

Rights: Copyright © 2010 Mathematical Society of Japan


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